The phase problem remains the major stumbling block of X-ray crystallography, and has constantly hampered the progress of our knowledge of the precise three-dimensional structure of biological macromolecules. We propose to strengthen the methodology of direct solution of the phase problem, so as to eventually bypass the use of isomorphous heavy-atom derivatives. For this purpose, we will undertake the elaboration of an algebraic formalism which constitutes the core of mathematical crystallography. This development will be directed towards two main goals: 1. The construction of eigenvector bases which diagonalize the Discrete Fourier Transform under general crystallographic symmetry. Such bases will afford considerable gains of efficiency in the simultaneous imposition of constraints in direct and in reciprocal space, a typical situation in the solution of the phase problem. 2. The consolidation of the algebraic machinery by which phase information available from probability calculations is collated and refined in existing direct methods. The efforts undertaken in the proposed research will be essentially synergistic with those directed by the Principal Investigator towards the development of a new phasing algorithm based on entropy maximization.